\(3\cos \theta + 4\sin \theta\) may be written in the form \(r\cos(\theta - \alpha)\), where \(r > 0\).
- Calculate the value of \(r\), and show that one value of \(\alpha\) is approximately \(53.1^\circ\).
Can we expand the expression \(\cos(\theta - \alpha)\)?
- Hence show that one solution of \(3\cos \theta + 4\sin \theta = 2\) is approximately \(120^\circ\).
Can we use the result from part (a)?
- State all the other solutions for \(0^\circ \leq \theta \leq 500^\circ.\)
How many solutions do you expect in the interval \(0\leq\theta<360^\circ\)?
How can we locate further solutions?
- Hence give the positive values of \(\theta\) less than \(500^\circ\) for which \(3\cos \theta + 4\sin \theta > 2\).
A sketch graph of \(y = 3\cos \theta + 4\sin \theta\) might be helpful. What happens at each of the solutions found in part (c)?